We prove that if $X$ is a smooth projective
threefold with $b_2=1$ and $Y$ is a Fano threefold
with $b_2=1$, then for a non-constant map
$f:X\rightarrow Y$, the degree of $f$ is bounded
in terms of the discrete invariants of $X$ and $Y$.
Also, we obtain some stronger restrictions on
maps between certain Fano threefolds.